Analytic feedback optimization for dipole fields

Abstract

Abstmcf-It has heeii sliowii that the method of using a siiiiple analytic model, such as A Fourier fuiictioii, witliiii t lie iterative op tiiiiizatioii cycle, for iiiagiietic field coiifiguratioiis which have aperture aspect ratios near iiiiity: (e.g. Quadrupoles) is very efiective [l]. The desired field coiifiguratioii is typically reachcd within two or three iteratioils. However for magnets with gap aspect ratios far from unity, such as iiiost dipoles, tlie Fourier fuiictioii is inappropriate. Here it is showii that a simpler iiiodel is effective for dipoles. It asstlilies that tlie field is proportional to the gap height. Then the field distribiitioii aiid the pole profile caii be inodeled nsiiig a polyimiiiid function, where for a uniforiii field, the terms liiglier tliaii the first are zero. Several examples are given for different types of dipole field coiifiguratioiis. I . ~N T R O D U C T I O NTlie standard procedure for optiiiiising t81ie field configrira,t<ioii of a magnets nw to va.ry oiie or more independent parameters uiit.il t,lie desired field configurat,ioii was a.cliieved. 'This u w hasically a. t.rial ancl error technique for wliicli t.lie number of it,erations u.as \.er!. tlrpeiideiit. upon the expericnc(> of die user. 'I'liis t.echnique coulcl soinet,iiiies be improved l~y usiug a Taylor scries fit between it,erat,ions. In gcweral t,liis ivorked fairl). ~v e l l for linear one dinieiisional prol~lenis. hlore recently ot,lier t,ecliniques sucli a.s neural networks aiid aiiiiealiiig E ? ] 1ial.e lieen tlcveloped, hut whilst. t,liese are po~verful tecliniyues for tlie general probleiii, t,liey are very slow t,o converge. Tlie niet,Iiod discxssed here uses expert. howledge of t,lie pa.rticu1a.r system being opt,imised. For iiiost iiiagiiet,ic field coiifigurat.ioiis a simple first order analytic solut,ion of tlie field configiiratioiis i, possible. It i s t.liis siinple ana.lytic solrit,ioii that, is Iiuilt iiito t h e feedhacli loop for the field opt!iiiiization t,ccliniciiie disci cl 1iei.c.. 111 t,lie case of a circular syinmet,ric syst,eiii whei it' magnet gap lias an aspect, ra.tio iiear iiiiit>y, siicli a.s a quailrupolc, a Fourier function can be used [I]. But, for magiiet,s wit,li aspect ra.t,ios far from unity suclr a.s iiiost dipole magnets t,lie fourier fuiict,ion is not, a suittable aiialyt,ic model. Here a even sinipler iiiodel ca.11 I I~ used, wliere i t , is assuiiiecl t,liat the field in the gap is proport8ioiia.l t o t,lie gap. The t,echiiique is t o start. wit81i a flat pole oft,lie desired. widt,Ii ant1 gap, then s o l i~ this using a tiiiit,e element program siicli as OPERA2 [ 3 ] . 'Typically (lie field will drol, of t,owartls t,lic: pole edges liecause of t , l l (~> ope11 Ellclh of the g a p Fol. t,he iiex t it era 1, i on t,lie g a p es a r e atl,just.ecl i i i iiiiwse proportioii to (,lie field cIi.011 ol' ( I ) > a.iitI aiiot,iier calcnla.t.ioii I 0 . 0 0 0 0.000 0.000 \ \ \ -0,0003--0.0004-I c o o r d 0.0 0 . 0 8.G 0.0 0 . 0 0 . 0 0.0 0 . 0 0.0 0.0 O.O\ 0.0 \ \ Fig. 1: ...